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cs-401r:homework-0.2 [2014/09/03 06:00] ringger created |
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- | <big>'''Bayes Nets'''</big> | + | = Bayes Nets = |

== Objectives == | == Objectives == | ||

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#* Assuming that the value of S is known, list ''all'' independence relations between T and other variables. Be sure to consider all of the cases in which the other variables in the model have known values and when they do not. | #* Assuming that the value of S is known, list ''all'' independence relations between T and other variables. Be sure to consider all of the cases in which the other variables in the model have known values and when they do not. | ||

# [10 points] Factor the joint distribution represented by the entire model shown in the figure according to the explicit independence assumptions represented in the model. | # [10 points] Factor the joint distribution represented by the entire model shown in the figure according to the explicit independence assumptions represented in the model. | ||

- | # [10 points] Write the four entries and their values in the conditional distribution for <math>P(L=0 | M=m,S=s)</math> (for m in {0,1} and s in {0,1}). | + | # [10 points] Write the four entries and their values in the conditional distribution for $P(L=0 | M=m,S=s)$ (for m in {0,1} and s in {0,1}). |

- | # [10 points] (a) Write an expression for the joint probability <math>P(T=1, R=0, L=0, M=0, S=1)</math> in terms of the probabilities given in the model (use the symbolic forms). (b) Then compute the actual probability. | + | # [10 points] (a) Write an expression for the joint probability $P(T=1, R=0, L=0, M=0, S=1)$ in terms of the probabilities given in the model (use the symbolic forms). (b) Then compute the actual probability. |

- | # [10 points] Compute <math>P(T=1, R=0, L=0)</math>. Show your work. | + | # [10 points] Compute $P(T=1, R=0, L=0)$. Show your work. |

- | # [10 points] Compute <math>P(T=1 | R=0, L=0)</math>. Show your work. | + | # [10 points] Compute $P(T=1 | R=0, L=0)$. Show your work. |

# [10 points] Prove that the relationship we call conditional independence is symmetric. Apply the same [[Proofs|standard of proof]] as in homework 0.1. | # [10 points] Prove that the relationship we call conditional independence is symmetric. Apply the same [[Proofs|standard of proof]] as in homework 0.1. | ||

- | #* In other words, show that (a) <math>P(X | Y, Z) = P(X | Z)</math> if and only if <math>P(Y | X, Z) = P(Y | Z)</math> | + | #* In other words, show that (a) $P(X | Y, Z) = P(X | Z)$ if and only if $P(Y | X, Z) = P(Y | Z)$ |

- | #* Equivalently, show that (b) <math>P(X, Y | Z) = P(X | Z) \cdot P(Y|Z)</math> if and only if <math>P(Y, X | Z) = P(Y | Z) \cdot P(X | Z)</math> | + | #* Equivalently, show that (b) $P(X, Y | Z) = P(X | Z) \cdot P(Y|Z)$ if and only if $P(Y, X | Z) = P(Y | Z) \cdot P(X | Z)$ |

- | #* (in other words, the "given <math>Z</math>" stays the same, while <math>X</math> and <math>Y</math> trade places). | + | #* (in other words, the "given $Z$" stays the same, while $X$ and $Y$ trade places). |

#* To be clear, prove (a) or (b). | #* To be clear, prove (a) or (b). | ||